WIMP Freeze-Out Relic Density Calculator

Introduction: Thermal Freeze-Out and the Abundance of Weakly Interacting Massive Particles

This WIMP freeze-out relic density calculator turns the thermal relic story into a quick estimate of Ω_χh² from a candidate's mass and annihilation cross-section. In the hot early universe, a WIMP stayed in equilibrium with the plasma, annihilating and being recreated as long as reactions were fast enough. As expansion cooled the bath and the equilibrium abundance fell away, the annihilation rate eventually could not keep up with the Hubble expansion rate. At that point chemical equilibrium broke down and the comoving density stopped tracking the equilibrium value, leaving a relic abundance that can survive to the present day. The calculator on this page condenses that picture into a simple interface, so you can test how a candidate's mass and ⟨σv⟩ feed into the present-day density parameter Ω_χh².

The freeze-out of a WIMP is governed by the Boltzmann equation for the number density n:

Here, H is the Hubble expansion rate, n_eq is the equilibrium number density, and ⟨σv⟩ denotes the thermally averaged annihilation cross-section times relative velocity. While solving this differential equation exactly requires numerical integration, analytic approximations capture the main dependence on particle physics parameters. We introduce the dimensionless quantity x = m_χ/T, where T is the temperature. Freeze-out occurs around x_f ≈ 20–30 for typical WIMPs. An iterative solution to the transcendental equation

provides a useful estimate. In this expression, g represents the internal degrees of freedom of the WIMP, g_* counts the effective number of relativistic degrees of freedom at freeze-out, and M_Pl is the Planck mass. After freeze-out, the comoving abundance remains constant. The resulting present-day relic density can be written in terms of the annihilation cross-section as

This inverse scaling is the reason the WIMP miracle is so attractive: a weak-scale annihilation cross-section naturally lands near the observed dark-matter abundance when the model is thermally produced. The calculator uses that scaling as its baseline. Enter m_χ in GeV and ⟨σv⟩ in cm³/s, and the tool solves for x_f, estimates T_f = m_χ/x_f, and evaluates Ω_χh². The classification then tells you whether the candidate is denser than, lighter than, or close to the roughly 0.12 cold-dark-matter benchmark.

To interpret the output, recall that cross-sections larger than 3×10⁻²⁶ cm³/s deplete the relic abundance, while smaller cross-sections leave too many WIMPs. In many models, the cross-section scales inversely with m_χ², but resonances, coannihilations, and threshold effects can complicate this picture. The calculator deliberately ignores these subtleties so the estimate stays simple and easy to compare across candidate masses and cross-sections. Nevertheless, the derived x_f and freeze-out temperature T_f = m_χ/x_f offer a reasonable indication of when in cosmic history freeze-out occurred. For example, a 100 GeV WIMP with the canonical cross-section freezes out at T_f ≈ 5 GeV, corresponding to microsecond-old universe times.

Understanding the freeze-out calculation provides insight into how relic densities depend on fundamental constants. The table below illustrates sample outputs for different masses and cross-sections:

These examples show how increasing the mass at fixed cross-section raises x_f slightly but leaves the relic density predominantly controlled by ⟨σv⟩. In realistic model-building, coannihilations with nearby states or velocity-dependent cross-sections modify the simple scaling. Nevertheless, the freeze-out paradigm serves as an instructive guide for evaluating hypothetical dark matter candidates across a wide parameter space.

The freeze-out story connects to several deep theoretical and observational efforts. On the theoretical side, supersymmetry, extra-dimensional models, and theories with hidden-sector gauge interactions all provide WIMP candidates whose masses and couplings can be plugged into this calculator. From the observational perspective, direct detection experiments measure the scattering cross-section of dark matter with nuclei, while indirect detection searches look for annihilation products such as gamma rays or antimatter. If a signal is observed, comparing the inferred annihilation cross-section with the relic density estimate offers a consistency check on whether the candidate was thermally produced. Likewise, collider experiments like the LHC can produce missing-energy events that might correspond to WIMPs; the calculator helps assess whether such particles could constitute all of dark matter.

Beyond the standard cosmological history, modifications such as an early period of matter domination, scalar-tensor gravity, or additional relativistic species (altering g_*) can shift the freeze-out abundance. Entropy injection after freeze-out can dilute relics, while non-thermal production channels can augment them. The calculator assumes none of these exotic scenarios, but the underlying formulae can be adapted by substituting the appropriate g_* or expansion history. This flexibility has encouraged a vast literature exploring non-standard freeze-out mechanisms like co-scattering, cannibalism, and bound-state effects, each of which leaves distinct imprints on the relic density.

For students learning cosmology, working through the derivation of the freeze-out equation offers a concrete application of thermodynamics in an expanding universe. Starting from the Boltzmann equation, one changes variables to the comoving yield Y = n/s, where s is the entropy density, and finds an equation of the form

with λ ∝ M_Pl m_χ ⟨σv⟩. Solving this equation yields the relic abundance. The calculator implicitly implements a simplified version of this solution, illustrating how microscopic physics feeds into macroscopic observables. Such exercises deepen intuition about the interplay between particle properties and cosmic evolution.

In summary, the WIMP freeze-out relic density calculator provides a quick yet informative window into one of the most influential ideas in dark matter physics. By entering just a mass and annihilation cross-section, users can estimate the abundance and assess whether a candidate aligns with cosmological observations. While full model assessments require more detailed computations, this tool highlights the parametric dependencies that underlie the celebrated WIMP miracle and continues to inspire experimental searches across the globe.

How to use this WIMP freeze-out calculator

1. Enter WIMP Mass m χ (GeV) for the candidate you want to test.
2. Enter Annihilation Cross-Section ⟨σv⟩ (cm³/s) for the same particle, using the unit shown in the field.
3. Run the freeze-out estimate and compare it with a second mass or cross-section choice before you rely on the relic-density result.

Formula: how the WIMP freeze-out estimate is built

The result can be read as the freeze-out estimate built from m χ and ⟨σv⟩, with xf solved iteratively and Ω_χh² then inferred from the benchmark inverse relation. Keep the mass in GeV and the annihilation cross-section in cm³/s exactly as requested by the form, because the calculation assumes consistent units throughout.

Worked example: compare one WIMP freeze-out scenario

A useful check is to hold the WIMP mass fixed and lower ⟨σv⟩; the calculator will return a larger Ω_χh² because annihilations stop earlier. Raise ⟨σv⟩ instead and the relic density falls, while x_f and T_f shift only modestly. That makes the cross-section the main lever to watch when you want to see whether the candidate looks overabundant, underabundant, or close to the dark-matter benchmark.

WIMP freeze-out limitations and assumptions

This tool uses the standard thermal freeze-out approximation, so it is best treated as a fast estimate rather than a full Boltzmann-solver output. The result depends on the mass and annihilation cross-section you enter, on the units staying consistent, and on the fixed choices of g = 2 and g* = 90 built into the calculator. It leaves out coannihilations, resonances, threshold effects, entropy injection, and non-standard expansion histories, so it should not be used in place of model-specific review or source data that can change over time.